Fixed Point Theorems and Applications
An element \(x \in X\) is called a fixed point of the mapping \(f:X \rightarrow X\) if \(f(x) = x\). Many problems, looking rather dissimilar at a first glance, from various domains of mathematics, can be reduced to the search for fixed points of appropriate mappings. For this reason each of the theorems on existence of fixed points discussed in the present chapter has numerous and often very elegant applications. Below we present three classical fix-point theorems: the Banach’s theorem on contractive mappings, the Schauder’s principle (which are applied for demonstrations of the Picard and Peano theorems on the existence of a solution to the Cauchy problem for differential equations and to the Lomonosov invariant subspace theorem), and the Kakutani’s theorem about common fixed points of a family of isometries, with application to the existence of a Haar measure on a compact topological group.